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midterm1_solutions

# midterm1_solutions - Problem 1[7 Points Consider the...

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Unformatted text preview: Problem 1 [7 Points] Consider the triangle R with vertices (—2, 2), (1, ~1), and (4, 2). (a) [2 Points] Sketch the triangle R. (b) [5 Points] Find the area of the triangle R using a double integral 4/6114. Problem 2 [8 Points] Consider the integral 3/1: )( 6*) :1: 2X (a) [2 Points] Sketch the region of integration. ‘g/Zéxe'l (b) [3 Points] Write an equivalent double integral With the order of integration reversed. ZX g S 6X20” oh 0 O (c) [3 Points] Use the equivalent double integral from part (b) to evaluate the above inte— gral. ‘u l ’2‘! .. .42 ~42 ’X “l O Q 0 O 0 : \——-L 9, Problem 3 [7 Points] An agricultural sprinkler distributes water in a circle of radius 100 feet. By placing a few random cans in this circle, it is determined that the sprinkler supplies water at a depth f (r) z 6—"2 feet of water at a distance of 7* feet from the sprinkler in 1 hour. (a) [5 Points] How much water does the sprinkler supply in 1 hour to the region within 100 feet of the sprinkler? (b) [2 Points] What is the average value of the function f over the region Within 100 feet of the sprinkler? «+2 SJ?“ OlA' —l00017‘ iréve §[JA l Trurooo E a: < % > Problem 4 [8 Points] Consider the region and the function - W (208(102 + y2). (a) [4 Points] Change the Cartesian integral 4/ m y) M into a polar integral. (b) [4 Points] Evaluate the polar integral. :IJ \’ C039 (0! l’z (ii—J9 Fr TVA S3? 9“ n— 2 A . - ’2, 2 Z :l 077 ﬁ— : ﬂuvial? = 1 qur2 : LWEJ-sms) 2 2 1'0 :“l \E 7E “L z \ 2/ 1...:- Problem 5 [10 Points] Let D be the tetrahedron in (cc, y, z)—space bounded by the four planes at = 0, y = 0, z z 0, and m+y+2= 1. (a) [2 Points] Sketch the tetrahedron D. /\ 7; 4g Kind-4 (b) [3 Points] Determine a description of D in terms of suitable bounds for x, y, and z. bl‘éLXﬁgﬁ) Oéx‘él ﬁbélrx) (b) [5 Points] Evaluate the integral l—x l—x-«n l-X S j S >< oh 0‘: 01" = [ng-x'nwlil 4" O 0 D o O \ l l : S X (0-x)2~(‘321))okx—l§(x-—szxx3)Jx 0 Z 0 \ L_Z—. L (0’9” .L T ELL 3 J“ Y) z .32 ‘2 l: Z‘f Problem 6 [10 Points] Consider the domain D={(x,y,z) I 09:31, 0:y:1~x, 03232} with constant density of mass (505,11, 2) 2 1 for all (rug/,2) E D. (a) [3 Points] Compute the mass M of the domain I). \ 2 \ex' '\ szjfgawzj f A A - 2 013‘? X- [Naddldx CD 0 D 0 O 0 \ ‘ \ t \, _ 1 ZS( YMXX Z(X'£)) 12([»i):l U L 2 0 «__._—p (b) [5 Points] Compute the moment of inertia ]y of the domain D about the y—axis. 'l—x 11-. (Hoar) 5‘ JV : J] {(64.22) 013 the w £5 0 0 O ‘ 2 2 X Shawn 21mm) oh Ax o o ' ~ 2‘ ,, 0(x : SLZXQ (I'¥)~r j:(\~x)>dx 2825;) (BX‘¥T)(\X) 0K 0 ' (—3x3+°>x2'*‘”‘“‘) * o W (c) [2 Points] Determine the radius of gyration 0f the domain D about the y-axis. n W ti ll NIX/v ...
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midterm1_solutions - Problem 1[7 Points Consider the...

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